Finite Group Databasebeta

← #13161253

Subgroups — #13161253

Order 1728.

768 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#947C211yesmaximal, characteristic
2#11723534C211yesmaximal, characteristic, complemented
3#11722205C211yesmaximal, characteristic, complemented
4C3×#63662C211yesmaximal, characteristic, complemented
5#11722032C211yesmaximal, characteristic
6#11722031C211yesmaximal, characteristic
7#11722019C211yesmaximal, characteristic, complemented
8#10592754—31nomaximal
9#10591624—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×#6293—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×(C2×D3)⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21C32×#124—31no—
22#64078—31no—
23C3×#947D314yes—
24#63918—34no—
25#63662—31no—
26#63662—34no—
27#63662—64no—
28#63662D311yescharacteristic, complemented
29#63659—31no—
30#63659—31no—
31#63630—31no—
32C3×#835—34no—
33#63563—34no—
34#63562—34no—
35#63554—34no—
36C22×C32×D3—21no—
37C33×Q8—21no—
38C33×Q8—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45C32×D12—21no—
46C32×D12—21no—
47#6293—42no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3×C3⋊(C3⋊C8)—21no—
50#5055—94no—
51#4794—91no—
52C2×C32×Q8—24no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×D4—31no—
56C2×#634—61no—
57C2×C3×D12C2×D314yes—
58C2×#627—64no—
59C32×C22⋊C4—61no—
60C32×C22⋊C4—61no—
61C32×C42—31no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—34no—
64C4⋊(C3⋊Dic3)—61no—
65C4⋊(C3⋊Dic3)—64no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×(C2×D3)⋊C4—24no—
72C3×(C2×D3)⋊C4—24no—
73C3×C4⋊Dic3—34no—
74C3×C4×Dic3C2×D314yes—
75C2×C3×C3⋊C8—34no—
76Dic3⋊Dic3—64no—
77C22×C33C2×D411yescharacteristic
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C2×C32×D3—41no—
81C33×C4—41no—
82C33×C4#4111yescharacteristic
83C33×C4#4111yescharacteristic
84C3⋊(C3⋊Dic3)—41no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—21no—
87C32×Dic3—21no—
88C3×#124—34no—
89#947C3⋊D311yescharacteristic
90#888—91no—
91#869—94no—
92C3×#103—91no—
93#835—34no—
94#835—94no—
95#835—91no—
96#835—184no—
97#832—94no—
98#832—94no—
99#821—91no—
100#815—91no—
101#809—91no—
102#803—94no—
103C23×C32—61no—
104C22×C3×D3—24no—
105C32×Q8—28no—
106C32×Q8—21no—
107C32×Q8—24no—
108C32×Q8—21no—
109C32×Q8—24no—
110C32×Q8—44no—
111C32×D4—61no—
112C32×D4—61no—
113C2×C32×C4—21no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—31no—
117C2×C32×C4—31no—
118C2×C32×C4—44no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—124no—
125#634—122no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3#6714yes—
128C2×C3×Dic3#6714yes—
129C3×D12—24no—
130C3×D12—24no—
131#627—128no—
132C3⋊(C3⋊C8)—21no—
133C3⋊(C3⋊C8)—61no—
134C3⋊(C3⋊C8)—64no—
135C3⋊(C3⋊C8)—124no—
136C3×C3⋊C8—64no—
137#489—271noSylow
138C2×C33#12411yescharacteristic
139C2×C33AGL(1,ℤ/8)11yescharacteristic
140C2×C33#13711yescharacteristic
141C32×D3—81no—
142C2×C3×Q8—24no—
143C2×C3×Q8D3214yes—
144C2×C3×Q8C2×C3⋊D311yescharacteristic
145C2×C3×D4—34no—
146C2×D12C2×C3⋊D311yescharacteristic
147C2×Dic6—184no—
148C2×Dic6—181no—
149C2×C3×C8—91no—
150C3×C4⋊C4—91no—
151C3×C22⋊C4—64no—
152C3×C22⋊C4—64no—
153C3×C42—34no—
154(C2×D3)⋊C4—21no—
155(C2×D3)⋊C4—21no—
156C4⋊Dic3—34no—
157C4⋊Dic3—94no—
158C4⋊Dic3—91no—
159C4⋊Dic3—184no—
160C4⋊Dic3—184no—
161Dic3⋊C4—181no—
162C4×Dic3C2×C3⋊D311yescharacteristic
163C2×C3⋊C8—34no—
164C2×C3⋊C8—94no—
165C2×C3⋊C8—91no—
166C2×C3⋊C8—184no—
167C22×C32—24no—
168C22×C32—121no—
169C22×C32—121no—
170C22×C32—121no—
171C22×C32D3×D411yescharacteristic
172C22×C32C2×#6714yes—
173C2×C3×D3—44no—
174C2×C3×D3—44no—
175C2×C3×D3—44no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—48no—
179C32×C4—41no—
180C32×C4—44no—
181C32×C4—61no—
182C32×C4—61no—
183C32×C4#23511yescharacteristic
184C32×C4#23714yes—
185C32×C4#23714yes—
186C32×C4#23711yescharacteristic
187C3⋊Dic3—41no—
188C3⋊Dic3—121no—
189C3⋊Dic3—124no—
190C3⋊Dic3—244no—
191C3×Dic3—24no—
192C3×Dic3—24no—
193C3×Dic3—124no—
194C4⋊Q8—271no—
195#124—31no—
196#103—91no—
197#103—271no—
198#102—271no—
199#102—271no—
200#97—271no—
201C33#48911yescharacteristic, complemented, Sylow
202C23×C3—64no—
203C22×D3—21no—
204C3×Q8—28no—
205C3×Q8—24no—
206C3×Q8—24no—
207C3×Q8—21no—
208C3×Q8—21no—
209C3×Q8—44no—
210C3×D4—64no—
211C3×D4—64no—
212C2×C3×C4—24no—
213C2×C3×C4—21no—
214C2×C3×C4—24no—
215C2×C3×C4—34no—
216C2×C3×C4—34no—
217C2×C3×C4—44no—
218C2×C3×C4—181no—
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—64no—
222C2×Dic3—184no—
223C2×Dic3—181no—
224C2×Dic3—364no—
225C2×Dic3#63811yescharacteristic
226C2×Dic3#63811yescharacteristic
227D12—21no—
228D12—21no—
229Dic6—362no—
230Dic6—368no—
231C3×C8—181no—
232C3⋊C8—64no—
233C3⋊C8—184no—
234C3⋊C8—181no—
235C3⋊C8—364no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—241no—
240C2×C32#88511yescharacteristic
241C2×C32#91411yescharacteristic
242C2×C32#91811yescharacteristic
243C2×C32#93614yes—
244C2×C32#94914yes—
245C2×C32#95114yes—
246C3×D3—84no—
247C2×Q8—541no—
248C2×Q8D3×C3⋊D311yescharacteristic
249C2×D4—31no—
250C2×C8—91no—
251C2×C8—271no—
252C4⋊C4—91no—
253C4⋊C4—271no—
254C4⋊C4—541no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C42—31no—
258C22×C3—24no—
259C22×C3—124no—
260C22×C3—124no—
261C22×C3—124no—
262C22×C3D3×#6714yes—
263C22×C3C2×#63811yescharacteristic
264C2×D3—41no—
265C2×D3—41no—
266C2×D3—41no—
267C3×C4—24no—
268C3×C4—24no—
269C3×C4—48no—
270C3×C4—41no—
271C3×C4—44no—
272C3×C4—64no—
273C3×C4—64no—
274C3×C4—361no—
275C3×C4#380414yes—
276C3×C4#380514yes—
277C3×C4#383011yescharacteristic
278C3×C4#383011yescharacteristic
279Dic3—21no—
280Dic3—21no—
281Dic3—124no—
282Dic3—361no—
283Dic3—364no—
284Dic3—724no—
285C32—24no—
286C32#479411yescharacteristic, complemented
287C32#505514yescomplemented
288C23—61no—
289Q8—21no—
290Q8—21no—
291Q8—1082no—
292D4—61no—
293D4—61no—
294C2×C4—21no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—181no—
298C2×C4—541no—
299C2×C4C2×D3×C3⋊D311yescharacteristic
300C8—181no—
301C8—541no—
302C2×C3—24no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—244no—
306C2×C3#6393914yes—
307C2×C3#6394314yes—
308C2×C3#6395914yes—
309C2×C3#6414511yescharacteristic
310C2×C3#6415811yescharacteristic
311C2×C3#6416011yescharacteristic
312D3—81no—
313C22—121no—
314C22—121no—
315C22—121no—
316C22D3×#63811yescharacteristic
317C4—41no—
318C4—61no—
319C4—61no—
320C4—361no—
321C4—1081no—
322C4#8902811yescharacteristic
323C4#8902911yescharacteristic
324C3—24no—
325C3#1059162414yescomplemented
326C3#1059275411yescharacteristic, complemented
327C2—241no—
328C2#1172356811yescharacteristic
329C2#1172357411yescharacteristic
330C2#1172360011yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#489
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11723534C2semidirect
3. #11722205C2semidirect
4. C3×#63662C2semidirect
7. #11722019C2semidirect
28. #63662D3semidirect
201. C33#489semidirect
286. C32#4794semidirect
287. C32#5055semidirect
325. C3#10591624semidirect
326. C3#10592754semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C32×#9472
#117235342
#117222052
C3×#636622
#117220322
#117220312
#117220192
#105927543
#105916243